Lot 9058, a damp-stained and mildewed medieval Greek liturgical book, went up for auction at Christie’s in New York on 29 October 1998. It was embroiled in controversy: the Greek Orthodox Patriarchate of Jerusalem had already brought an injunction against Christie’s, claiming the book had been stolen from a monastery in Constantinople during the collapse of the Ottoman Empire. But the court ruled in Christie’s favour: the book’s owner was French, and under French law her father had obtained it in good faith. The auction went ahead.
It wasn’t simply a liturgical book. Beneath the red ink of its prayers was a fainter text in a crabbed perpendicular hand: Method of Mechanical Theorems, attributed to the Greek inventor and mathematician Archimedes, which had fallen out of circulation by the 13th century. In 1906, before the manuscript disappeared from Constantinople, the Danish historian J.L. Heiburg noticed the darker text and took some grainy photographs, from which he copied what he could see of the Greek through a magnifying glass. But here was the palimpsest itself, finally available for further study. The auction was fraught. Major American universities competed with representatives of the patriarchate in a strange clash of secular and spiritual learning. In the end, an anonymous buyer swooped in with a bid of $2 million. Historians worried that the palimpsest would never be seen in public again. But the buyer surprised everyone; less than a year later, it was on long-term loan to the Walters Art Museum in Baltimore. Since then, it has undergone a series of expensive interventions to repair surface damage and provide detailed digital scans, improving the reliability of the legible text.
Method of Mechanical Theorems is a letter from Archimedes to Eratosthenes, the head of the library of Alexandria and a renowned philosopher in his own right. Most of it is devoted to proving relationships between the centres of mass of different shapes. Archimedes starts with the familiar – parabolas, triangles, spheres – but quickly moves beyond them. As the historian of maths Reviel Netz puts it, one example features a shape like a ‘lipstick tip’ and another a ‘box bounded by eight symmetrically curved surfaces’. As in most Greek proofs, Archimedes expresses the area or volume of one thing in relation to another: this parabola is four-thirds as big as this triangle; that triangular prism is the same size as that lipstick tip.
Archimedes’ other works, many of which survive, also deal with theoretical geometry. Some concern magnitude, scale and approximation – for example, Sand Reckoner, in which he estimates how many grains of sand it would take to fill the universe. Others group extremely difficult geometrical proofs on specific themes. The most famous is The Sphere and the Cylinder, in which he proves that if you fit a sphere inside a cylinder so that it touches both sides, the sphere’s volume will be two-thirds that of the cylinder. But in Method of Mechanical Theorems Archimedes makes a playful confession. He has been using a ‘mechanical’ method of his own devising to formulate new mathematical problems. Once he has found a problem to work on, the mechanical method can also help him solve it. He encourages Eratosthenes to try it, speculating that others may find exciting new mathematics with it. When Heiburg’s transcription was circulated, it seemed to confirm a feature of Archimedes’ work that had been suspected for some time: some of his abstruse theories, particularly those about centres of mass, may have been scoped out using physical models made of wood and metal. Rather than relaying complex geometrical relationships from his mind’s eye onto the page, Archimedes had been patiently constructing, adjusting and balancing his models, and then doing the geometry to shore up his discoveries. The mechanical method exposed a way of thinking that set him apart from an earlier generation of philosopher-mathematicians: he trusted that the intellectual world of mathematics corresponded to material reality.
As Nicholas Nicastro points out in his biography, the stories that circulate about Archimedes aren’t necessarily reliable. In 212 bce, the Romans attacked his home town, Syracuse, one of the largest and wealthiest cities in the Mediterranean. Syracuse had long been an ally of Rome, but after Hannibal’s journey over the Alps its teenage ruler, Hieronymus, was tempted into a treaty with the Carthaginians. Rome responded swiftly. Archimedes, then in his seventies, was killed in the chaos. There are various accounts of his death, but one has gained more traction than the others: Archimedes was too busy drawing geometrical diagrams in the sand to look at the Roman soldier who demanded his surrender. The story tempts us to view him as a grandfatherly eccentric, a sort of proto-Einstein. But Nicastro notes that we have no good reason to accept it over other accounts. Perhaps Archimedes was fleeing with a chest of scientific instruments that were mistaken for treasure, or perhaps the Roman soldiers knew who he was and wanted to eliminate him for his role in the defence of the city. He was, after all, an engineer working in military service during a major war.
The other stories about Archimedes’ inventions point to his fascination with technology’s potential as well as his need to show off. He is said to have pulled a ship along the beach at Syracuse single-handedly, using a complex mechanical contraption to demonstrate the absurd multiplicative power of pulleys. According to Anthemius of Tralles, one of the architects of the Hagia Sophia, Archimedes found a way to set Roman ships on fire from a distance, harnessing and focusing the rays of the sun with hinged mirrors. The TV show MythBusters devoted two episodes to the claim in the mid-2000s, concluding – to the relief of parents everywhere – that it is more or less impossible to focus an ordinary mirror in this way. (Anthemius may have been influenced by the concerns of his own day: Byzantine optical science was obsessed with mirrors.) The Greek historian Polybius, born not long after the siege of Syracuse, writes that Archimedes constructed an enormous metal claw attached to a crane which swung out from the harbour walls, fastened into the hulls of Roman ships and toppled them, soldiers and all. It sounds like a Heath Robinson machine, but it might have worked. A recreation in an episode of Secrets of the Ancients demonstrated that if the claw was used to create latitudinal roll it could easily destabilise overloaded ships.
In the Hellenistic world of Archimedes’ birth, intellectuals weren’t meant to be practical. Stories of their childlike innocence were common. Some of the earliest concerned Thales, an astronomer in the sixth century bce who fell down a well while looking at the stars. The prejudice was partly just ordinary people poking fun at the pointy-heads: they might have brains, but we have common sense. But elite men in the classical Greek world distrusted technical knowledge, which they associated with slavery and femininity. Weaving and textile work was carried out almost entirely by women, and pottery, metalwork and toolmaking were the domain of slaves. If intellectuals knew about crafts, it was usually only through the businesses they owned. The orator Lysias owned a shield factory in which more than a hundred enslaved men worked; Demosthenes’ father had a sword-making business and a furniture workshop.
Socrates’ pupil Xenophon wrote in the fourth century bce that the trades of craftsmen were ‘held in low esteem in the cities’ because they ‘softened’ the bodies and souls of their practitioners and withered their social and political skills. Xenophon noted with approval that in the aristocratic and militaristic culture of Sparta, full citizens were forbidden from pursuing banausikai technai, the ‘mundane’ crafts of artisans; the enslaved population, known as the helots, produced everything that wasn’t imported from subordinate cities. Plato deplored trade of any kind, which he believed invited moral decline through greed: the most degenerate place, he argued, was a city that had grown rich producing goods for export. Greek myths about craftsmen tend to be cautionary. Hephaestus, the god of bronzeworkers, is lame and mocked for it; the titan Prometheus is punished for teaching technical skills to men; and the canniest craftsman of all, Daedalus, loses his son Icarus through his hubristic invention of artificial wings.
But Archimedes was a new type, a scientist who combined theory with practical know-how. The most famous story about him displays this combination. The ruler of Syracuse, a popular tyrant called Hieron II (the grandfather of the ruler who incurred Rome’s wrath), suspected he was being duped about the purity of a gold crown he had ordered as a religious offering. But he wasn’t certain – the crown did seem to weigh the right amount. Perplexed, he called in Archimedes. The rest is legend: Archimedes ran naked from his bath crying ‘eureka’, ‘I’ve found it’ (the Roman architect Vitruvius, who tells us this story, takes care to mention that he shouted in Greek). Archimedes’ realisation, based on the water that spilled out of the bath when he got in, was that he could measure the water displaced by the crown and compare it to the amount displaced by a piece of gold of equal weight. Since gold is very dense, if the crown displaced more water than the equivalent piece of pure gold, it must have been adulterated with some lighter metal. This might seem at first glance like another homespun hagiography. But as Nicastro points out, something about it rings true. We see exactly the same principles Archimedes outlines in Method of Mechanical Theorems: the close relationship between reality and mathematics, and the use of an experiment to solve a geometrical problem – in this case, the volume of an irregular solid. (The goldsmith had tried to deceive Hieron.)
For Nicastro, Archimedes was a true scientist who wouldn’t have been caught dead endorsing the ‘quaint beliefs’ held by other Greeks: that circles are the most perfect shape, or that objects fall to earth seeking their proper place. He invokes Leonardo da Vinci and Galileo in support: they recognised Archimedes as a fellow genius, so we should too. The problem with this argument is that it rests on a sense of mathematical physics as a special lineage, which takes us on, presumably, to Tesla, Feynman and Hawking. It’s true that Leonardo and Galileo, trained on a tangled web of Aristotelian, Ptolemaic and Galenic scholasticism, might have found Archimedes refreshing and identified him as a kindred spirit. But what makes Archimedes interesting is not where he comes in a family tree of genius, but something specific to his intellectual approach: the way in which his understanding of the mathematical world diverged so intriguingly from that of his predecessors.
The earliest Greek mathematicians had studied the properties of numbers and of ordinary shapes, as had the Mesopotamians and Egyptians before them. Inspired by them, and heeding the doubts expressed by Parmenides and Zeno about the reliability of the senses, Plato distrusted material reality and accorded a special status to mathematics. For him, it occupied a middle position between the divine realm of the forms and the imperfection of the material world. Where we couldn’t access truth directly through perception, maths was the next best thing. Aristotle reacted against Plato’s approach. He was a specimen collector and dissector at heart, and his philosophical system was based on his love of the physical stuff of the world. Mathematics was simply a tool, and Aristotle was never entirely comfortable with it. Most thinking about maths since has taken one of these two paths: it’s either a portal to a higher reality or a convenient way to model the world around us.
Archimedes took a different tack. For him, the mathematical and physical worlds were as real as each other and mutually reinforcing. To know something about the physical world, you could use maths to model it. But more radically, you could also use the physical world to open up and model mathematical ideas: balances, levers and floating objects could be treated as geometrical proofs. This approach has never had the dominance of the other two – though traces of Method of Mechanical Theorems can be seen when modern mathematicians use computer simulations to inspire their conjectures.
Method was probably among the works of Archimedes collected by Isidore of Miletus in the sixth century ce. From there, it made its way into the scholarly penumbra of the Byzantine establishment; the underlying text in the palimpsest may have been copied out in the ninth century by Leo the Geometer, a teacher and cousin of John VII, patriarch of Constantinople, who was fascinated by Archimedes. But then the thread breaks: Constantinople was sacked in 1204 during the Fourth Crusade, leaving the city and its schools in disarray. The liturgical palimpsest was made in the wake of these events, possibly in Jerusalem, from a selection of extraneous Byzantine manuscripts, hastily bundled together. Parchment was expensive and in short supply, and whoever wrote the prayers probably had no interest in the ghostly letters beneath.